One takeaway message from our results is that when showing space lower bounds for data stream algorithms computing symmetric functions on product distributions, by looking at 2-player communication complexity (which is by far the most common communication setup), there is only an \(O(1)\)  factor loss for error probability \(1/\mathsf {poly}(k)\)  protocols, and an \({{O} \left( {\log k} \right)}\)  factor loss for constant error protocols.
However, for non-product distributions, which are often needed to show hardness of approximation in data streams (such as for the frequency moments [1]}), one may need to use as many as \(k\)  players in order to obtain a non-trivial lower bound from communication complexity.
